The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the "Segre map") of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced which establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semitopological K-theory and an interpretation of the "topological filtration" on singular cohomology groups in K- theoretic terms.

Abstract. In this paper, we introduce the “semi-topological K-homology ” of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological K-homology of the associated analytic space Y an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the “Bott inverted ” semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X an. In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason’s celebrated theorem that “Bott inverted” algebraic K-theory with Z/n coefficients coincides with topological K-theory with Z/n coefficients. 1.

We establish the existence of an “Atiyah-Hirzebruch-like” spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that connects the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) cohomology of complex varieties introduced by H. Gillet and C. Soulé — to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also

Abstract. We establish the analogue of the Friedlander-Mazur conjecture for Teh’s reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety X vanishes in homological degrees larger than the dimension of X in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of dos Santos ’ real Lawson homology as the homotopy groups of the topological group of averaged cycles. We also establish an equivariant Poincare duality between equivariant Friedlander-Walker real morphic cohomology and dos Santos ’ real Lawson homology. We use this together with an equivariant extension of the mod-2 Beilinson-Lichtenbaum conjecture to compute some real

Abstract. Let E be a row-finite quiver and let E0 be the set of vertices of E; consider the adjacency matrix N ′ E = (nij) ∈ Z (E0×E0), nij = # { arrows from i to j}. Write Nt E and 1 for the matrices ∈ Z(E0×E0\Sink(E)) which result from N ′t E and from the identity matrix after removing the columns corresponding to sinks. We consider the K-theory of the Leavitt algebra LR(E) = LZ(E) ⊗ R. We show that if R is either a Noetherian regular ring or a stable C∗-algebra, then there is an exact sequence (n ∈ Z)